Algebra

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Week 1
(Section 6.1 – 6.4, 6.6 – 6.8)

Dr. Dutta

Ashford University

Definition

A rational expression is of the form: or

Example:

If the value of a rational expression is used to find the value of a second variable, then we have a rational function

Example: y = .

For x = 1, we can find the value of y as y =

Quick Review – GCF and Reducing to Lowest Term

In MAT 221, we learned about reducing to lowest terms and greatest common factor. Here is a quick Review:

Reducing to lowest term means the fraction (or rational expression) should be in the simplest form; that is the numerator (top) and the denominator (bottom) should not have any common factors.
Example: is not in the simple form (lowest form) because 20 and 25 have

common factors of 5. Thus, reducing to lowest form: . Thus, 4/5 is the lowest form of 20/25.

Greatest Common factor is the largest integer that is the factor of two or more integers. Example: 6 is the GCF of 12 and 18 because

2 is a factor of 12 and 18

3 is a factor of 12 and 18

4 is a factor of 12 and 18

6 is a factor of 12 and 18.

Among all these factors, {2, 3, 4, 6}, 6 is the largest and thus it is the GCF of 12 and 18.

Reducing Rational Expressions

In order to reduce a rational expression, we follow the following two rules:

1. Factor the numerator and denominator

2. Divide the numerator and denominator by GCF

Example:

Step 1: Factor the numerator and denominator.

Step 2: Note that the numerator has the expression x + 2y; the denominator also has the

expression x + 2y. Whenever you have same expression in the numerator and

denominator, cancel them out, and thus we are left with the final answer as

Another Example

Example: Reduce to lowest term:

Step 1: Factor out 2 from the numerator and 3 from the denominator:

Step 2: Recall factorization of quadratic equation from MAT 221 and the formula (x – y)(x + y) = x2 – y2. So, apply the factorization technique of quadratic equations to write the numerator as (x + 2)(x + 3) and the denominator as (x + 3)(x – 3) and thus the expression becomes:

Step 3: Cancel out the common factors present in both numerator and denominator. In this example, we see that x + 3 is present in both numerator and denominator. Thus, cancelling that out, we get the final answer as 2(x + 2)/3(x – 3):

Domain

Domain – it is all the possible input values that allows a function to work.
For a rational function, denominator cannot have zeros because ,where a can be any number or algebraic expression, is undefined in Mathematics.

Find the domain

Solution: the function is a rational expression. The rule says that that denominator

cannot have zero. So, we need to find out if and when the denominator can become

zero.

Step 1: Find where the denominator become zero by solving the denominator: x + 5 = 0

x = -5

At x = -5, the denominator becomes zero and thus we cannot have this value.

Thus, the domain of this rational expression is all numbers except x = -5.

Another Example of Domain

Find the domain of

Step 1: Set the denominator equal to zero and solve for x:

x2 + x – 12 = 0

(x + 4)(x -3) = 0

x = -4 and x = 3

Step 2: Write the conclusion: thus, the domain should exclude those values of x. But it can have all the other values including any x-value lying between -4 and 3

Answer: the domain is any number except x = -4 and x = 3.

Multiplication and Division of Rational Expressions

To multiply rational expressions, remember the following rule, that is multiply the numerators and the denominators:

To divide rational expressions, remember the following rule, that flip the denominator and multiply it to the numerator:

Examples of Multiplication of Rational Expressions

Step 1: Multiply the numerator and denominator
Step 2: Simplify the expression through factorization and cancellation

Multiply

Step 1: Multiply top and bottom:

Step 2: Do simplification through cancellation:

, because x from the denominator cancel out a x

from the numerator and we are left with 5 x’s in the top:

, because 2a’s from top cancels out 2a’s from bottom

and we are left with three a’s in the bottom.

, cancel out the GCF of 42 and 35, which is 7.

Another Example of Rational Expression

Multiply

You do not have to all the time follow the two rules of multiplying top and bottom and

doing simplification in that order. You can first do simplification and then multiply the

top and bottom. That’s what we are going to do in the above example. In order to do

simplification, always see if you can factor things out, find GCF, etc.

Step 1: 2x + 2y has the common factor of 2; 6x + 6y has the common factor of 6. Thus,

factor these numbers out:

Step 2: x + y is present in both numerator and denominator. Cancel them out to get:

Step 3: 6 and 15 has common factors of 3. Thus, factor that out, and cancel out all the

common factors to get the final result of 5/7.

Division of Rational Expressions

For division, always apply the first rule, that is (a/b)/(c/d) = (a/b)*(d/c). Without this first step, you cannot perform the division of rational expressions.

Example: Do

Step 1: Flip and multiply:

Step 2: Do the simplification: , because 2 cancels out 4; x cancels out one of

the x’s in the numerator.

Final answer is x/2

Another Example of Rational Expression Division

Step 1: Flip and multiply:

Step 2: Factor and simplify: , because by applying

the knowledge of quadratic equations we know that x2 + 2x + 1 = (x + 1)2 and that x2

1 = (x + 1)(x – 1)

Since x + 1 is present in both top and bottom, we can cancel out to get:

Step 3: Do the multiplication and write it together as

Addition and Subtraction of Rational Expressions

Here are the rules:

If the denominator is same, then you just add the top and the denominator remains as it

is:

If the denominators are not same, then (a). You need to factor out the denominators, if

factorization required; (b). if numerator and denominator can be simplified, then that

needs to be done before addition/subtraction; (c). then the expressions are

added/subtracted by doing algebraic manipulations so that the denominator of each

expression is same and you can apply the formula

Examples

Add/subtract:

NOTE that the denominators are same. So, just add the top. Also, realize that we add

rational expressions just like we will add fractions in arithmetic:

So, by the rule:

Now, step 2 and that is simplify the expression as much as possible:

Example of Addition/Subtraction

Add:

NOTE that the denominators are not same. Thus, you cannot simply add them.

Step 1: Factorize the denominator. x2 – 9 = (x - 3)(x + 3) and x2 + 4x + 3 = (x + 1)(x + 3) to

get

Step 2: In this example, no further simplification of numerator and denominator can be

done. Thus, we continue to the process of addition. To add these types of expressions, the

simplest approach is this.

Look at the first expression, and compare it’s denominator with that of the second

Expression. It has x+ 3; the second expression also has x + 3; it has x – 3; the second

expression does not have that. So, we are going to multiply the top and bottom of the

second expression with x – 3.

Now do the same thing with the second expression. Compare its denominator to that of the

first one. Both of have x + 3. however, second one has x + 1 which is missing in the first

one. So, we are going to multiply the top and bottom of the 1st with x + 1

The Previous Example Continues

Step 2 continues: Thus, 1st expression multiplied with x + 1; the 2nd one with x - 3:

Step 3: VOILA! Now check out their denominators. Each has same denominator and

thus we can just add the top:

Step 4: Simplify and get the final answer:

Steps To Solve Rational Expressions

Make the denominator of each quotient same.
Cross-multiply/divide/add/subtract
Solve

Example

Solve

The denominator consists of 4 and 3. The smallest number divisible by 4 and 3 is 12 (the least common denominator is 12). Thus, multiply the whole expression, both left and right, by 12:

Start solving:

Multiplying both sides of –x = 14 by -1 to get x = -14

Example

Solve

Denominators are x, 3x.

The least common denominator is then 3x

Multiply the whole given equation by 3x:

Start solving:

Example

Solve:

The denominators are 2x, 9, 18 and 3x

The lowest common denominator is 18x

Multiply the equation by 18x:

Previous Example Continues

Start Solving:

Example

Solve:

Step 1: Denominators on the left are x+3 and x – 2

Step 2: Multiply the top and bottom of with x – 2

And multiply the top and bottom of with x + 3 to get:

Previous Example Continues

Step 3: The denominators on the left are same, thus combine the left

side:

Step 4: Is the denominator on the right same as that of in the left?

Two ways to answer: 1. Factorize the quadratic equation in the right or

2. Distribute (FOIL) the expression (x+3)(x-2) in the left:

x2 + x – 6 = (x+3)(x-2).

Yes, they are same.

Previous Example Continues

Step 5: Since the denominator in the both left and right are same,

therefore we can ignore the denominators and just focus on the

numerator because math says if , then a = c

Thus, x – 27 = -20

 x = -20 + 27 = 7 is the solution

An Important Concept

In Math, you cannot have a zero in the denominator because dividing anything by zero does not make sense.
Thus, whenever we are dealing with rational functions (equations), sometimes we might be asked to at what points the denominator become zero; or sometimes, we might have to pay attention to that fact while solving the equation.
So, how do we answer the question:

Set the denominator equal to zero and solve for the value of x.

Example: Let the equation be . Find the values of

x for which the denominator is zero. This problem is very similar to finding the domain of

rational expression.

Solution

We know that when we manipulate the example, we get:

Since the denominator is (x+3)(x-2), set that equal to zero. Thus:

(x+3)(x-2) = 0  x+ 3 = 0 and x – 2 = 0  x = -3 and x = 2 are

the values of x for which the denominator will become zero.

Proportion

If a/b = c/d are two equal ratios, then the statement

is known as a proportion.

Property of Proportions

If , then ad = bc, that is product of numerator of first ratio and
denominator of second ratio is equal to the product of numerator of second ratio and denominator of first ratio.

Example:

Use of Property of Proportions

We use the property of proportions to solve simple equations.
Example 1 Solve for x:

Step 1: Apply the property: 4*x = 3*5  4x = 15 (multiplying both sides by 4 and then multiplying both sides by 5).
Step 2: Multiply both sides by reciprocal of 4:

Step 3: Write down the answer: x = 15/4 or or 3.75

Application of Proportion

A woman drives her car 235 miles in 5 hours. At this rate, how
far will she travel in 7 hours?
Step 1: Convert the word-problem into a proportion:
235 miles to 5 hours = how far to 7 hours

Step 2: Represent the unknown quantity with a variable x:

Step 3: Solve for x  235*7 = 5*x  1645 = 5x
Multiplying both sides by 1/5 (which is same as dividing both
sides by 5):  329 = x

Step 4: The woman will travel 329 miles in 7 hours

The End

So, in chapter 6 we learned about the following important concepts

Reducing a rational expression to lowest term

Finding the domain of a rational expression

Multiplication, division, addition and subtraction of rational expressions

Solving equations involving rational expressions

Ratio/Proportion and Application of Proportion to solve real-life problems.

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